Optimal control problems governed by a class of elliptic variational inequalities of the second kind are investigated. Applications include the optimal control of simplified friction, visco-plastic fluid flow and thin plate deformation. Based on a Tikhonov regularization of the dual problem, a family of primal-dual regularized control problems is introduced and strong convergence
of the regularized solutions towards the solution of the original control problem is verified. For each regularized problem an optimality system is derived and an optimality condition for the original control problem is obtained as limit of the regularized optimality systems. The resulting optimality system includes complementarity relations between the variables involved. Since the regularized optimality systems involve Newton differentiable functions, a semi-smooth Newton algorithm is proposed and its numerical performance investigated.